We study resonance asymptotics of Schroedinger operators in the scattering system consisting of several halflines attached to a compact metric graph or two or three dimensional Riemannian manifold. We are interested in the High energy behaviour of the number of resonances enclosed in the circle of radius R in the k-plane.

Davies and Pushnitski showed that there is a class of quantum graphs for which the constant by the leading term of Weyl asymptotics is smaller than the sum of the lenghts of the internal edges. We find a general criterion for this «non-Weyl» behaviour and explain how such graphs can be constructed. Furthermore, we show how the asymptotics may change in the presence of magnetic field. On the other hand, at least for certain classes of hedgehog manifolds different dimensionality of the scattering center and the leads ensures that this behaviour does not appear.

The talk is based on joint papers with P. Exner and E.B. Davies:

[1] E.B. Davies, P. Exner, J. Lipovsky: Non-Weyl asymptotics for quantum graphs with general coupling conditions, J. Phys. A43 (2010), 474013.

[2] P. Exner, J. Lipovsky: Non-Weyl resonance asymptotics for quantum graphs in a magnetic field, Phys. Lett. A375 (2011), 805-807.